Diophantine equation 26

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$${Published}$$  $${Online}$$  $${First}$$  $${(16/2/2024)}$$
$${Latest}$$  $${additions}$$  $${(16/2/2024)}$$
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$${Diophantine}$$ $${equation}$$ $${26}$$
$${(26.1)}$$  $${~~~\sum\limits_{i=1}^3\ a_i^6}$$ $${=}$$ $${\sum\limits_{j=1}^3\ b_j^6}$$
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$${case(26.1)}$$  $${~~~\sum\limits_{i=1}^3\ a_i^6}$$ $${=}$$ $${\sum\limits_{j=1}^3\ b_j^6}$$

$${Brudno, Delorme}$$による解
$${For~k = 2,6,}$$
$${a_1=(-n^4-n^3-5n^2+8n+8)\\a_2=(n^3+7n-2)(n+2)\\a_3=(9n^2+6n+12)\\b_1=(n^2-n+3)(n+2)^2\\b_2=(-4n^3-5n^2-8n+8)\\b_3=(-n^4+n^2+14n+4)}$$
この時、$${k=2,6}$$で下記方程式を満たす。
$${~~~\sum\limits_{i=1}^3\ a_i^k}$$ $${=}$$ $${\sum\limits_{j=1}^3\ b_j^k}$$

$${n=3}$$→
$${(a_1,a_2,a_3)=(121,230,111)}$$
$${(b_1,b_2,b_3)=(225,169,26)}$$
$${121^2+230^2+111^2=225^2+169^2+26^2}$$
$${121^6+230^6+111^6=225^6+169^6+26^6}$$
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$${REFERENCES}$$
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$${\small【Anthony~L.~Hart】}$$
$${\small 「A~Collection~of~Algebraic~Identities」}$$
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【目次001】・【目次002】・【目次003】
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